For the CMUT device, we started the design with the analytical modelling of the device and then FEM simulations were performed. The resonance frequency response of the CMUT device depend upon the device parameters and material properties. The resonance frequency is calculated by formula given in Eq. 1, and is found to be 4.1 MHz [14].

$${\text{f}}_{\text{r}}= \frac{0.47 \text{h}}{{\text{a}}^{2}}\sqrt{\frac{\text{E}}{{\rho }(1-{{\nu }}^{2})}}$$

1........... for a CMUT device is directly proportional to the thickness of the membrane and Young's modulus of the material. However, it is inversely proportional to the radius of the membrane, Poisson’s ratio, and density of the material.

For the transmission mode operation of the device, the DC voltage is applied between two electrodes, which leads to the generation of electrostatic force between the plates of the device defined by Eq. 2.$${\text{F}}*{\text{e}\text{l}\text{e}\text{c}\text{t}\text{r}\text{o}\text{s}\text{t}\text{a}\text{t}\text{i}\text{c}}= \left[\frac{{\pi }{\text{a}}^{2}}{2{\left({\text{d}}*{0}-\text{w}\right)}^{2}}+\frac{\text{a}}{\left({\text{d}}*{0}-\text{w}\right)}-1.918\right]{{\epsilon }}*{0}{\text{V}}^{2}$$

$${\text{P}}*{\text{e}\text{l}\text{e}\text{c}\text{t}\text{r}\text{o}\text{s}\text{t}\text{a}\text{t}\text{i}\text{c}}= {{\epsilon }}*{0}{\text{V}}^{2}\left[\frac{1}{2{\text{d}}*{0}^{2}}+\frac{1}{{\pi }\text{a}{\text{d}}*{0}}-\frac{1.918}{{\pi }{\text{a}}^{2}}\right]-{{\epsilon }}*{0}{\text{V}}^{2}\left[\frac{1}{{\text{d}}*{0}^{3}}+\frac{1}{{\pi }\text{a}{\text{d}}*{0}^{2}}\right]{\text{w}}*{0}$$

To compensate for the effect of electrostatic pressure, a reverse pressure known as elastic restoring pressure is also generated inside the membrane in the direction opposite to the electrostatic pressure; it is defined as [

15].$${\text{P}}*{\text{e}\text{l}\text{a}\text{s}\text{t}\text{i}\text{c}}= \left[\frac{4{\sigma }\text{h}}{{\text{a}}^{2}}+\frac{64\text{D}}{{\text{a}}^{4}}\right]{\text{w}}*{0}+\left[\frac{128{\alpha }\text{D}}{{\text{h}}^{2}{\text{a}}^{4}}\right]{\text{w}}_{0}^{3}$$

denotes the combined stiffness due to bending and residual stress, and second term defines the stiffness of the membrane due to non-linear stretching.

For the parallel plate actuator, distance travelled by diaphragm (membrane) for pull-in to occur is equal to one third of the total gap between the two plates of the actuator$${\text{w}}*{0-\text{P}\text{I}}= \frac{{\text{d}}*{0}}{3}$$

By substituting Eq. (5) in Eq. (3) and Eq. (4), we got the following two important equations:$${\text{P}}*{\text{P}\text{I}-\text{e}\text{l}\text{e}\text{c}\text{t}\text{r}\text{o}\text{s}\text{t}\text{a}\text{t}\text{i}\text{c}}= {{\epsilon }}*{0}{\text{V}}^{2}\left[\frac{1}{2{\text{d}}*{0}^{2}}+\frac{1}{{\pi }\text{a}{\text{d}}*{0}}-\frac{1.918}{{\pi }{\text{a}}^{2}}\right]-{{\epsilon }}*{0}{\text{V}}^{2}\left[\frac{1}{{\text{d}}*{0}^{3}}+\frac{1}{{\pi }\text{a}{\text{d}}*{0}^{2}}\right]\left[\frac{{\text{d}}*{0}}{3}\right]$$

$${\text{P}}*{\text{P}\text{I}-\text{e}\text{l}\text{a}\text{s}\text{t}\text{i}\text{c}}= \left[\frac{4{\sigma }\text{h}}{{\text{a}}^{2}}+\frac{64\text{D}}{{\text{a}}^{4}}\right]\left[\frac{{\text{d}}*{0}}{3}\right]+\left[\frac{128{\alpha }\text{D}}{{\text{h}}^{2}{\text{a}}^{4}}\right]{\left[\frac{{\text{d}}_{0}}{3}\right]}^{3}$$

By using the above two equations (6) and (7), we calculate the pull-in voltage for the membrane. The pull-in voltage is defined as the voltage value at which the membrane collapses at bottom electrode. The collapse voltage determined by Eq. (8) is 80.09 V.$${\text{V}}*{\text{P}\text{I}}= \sqrt{\frac{\left[\frac{64\text{D}}{{\text{a}}^{4}}+\frac{4{\sigma }\text{h}}{{\text{a}}^{2}}\right]\left(\frac{{\text{d}}*{0}}{3}\right)+\frac{128{\alpha }\text{D}}{{\text{h}}^{2}{\text{a}}^{4}}{\left(\frac{{\text{d}}*{0}}{3}\right)}^{3}}{{{\epsilon }}*{0}\left[\frac{1}{6{\text{d}}*{0}^{2}}+\frac{2}{3{\pi }\text{a}{\text{d}}*{0}}-\frac{1.918}{{\pi }{\text{a}}^{2}}\right]}}$$